| L(s) = 1 | + (2.17 − 1.39i)2-s + (−0.142 + 0.989i)3-s + (1.94 − 4.26i)4-s + (0.959 − 0.281i)5-s + (1.07 + 2.35i)6-s + (−1.28 − 1.48i)7-s + (−0.987 − 6.86i)8-s + (−0.959 − 0.281i)9-s + (1.69 − 1.95i)10-s + (3.45 + 2.21i)11-s + (3.93 + 2.53i)12-s + (−2.89 + 3.34i)13-s + (−4.87 − 1.43i)14-s + (0.142 + 0.989i)15-s + (−5.61 − 6.47i)16-s + (−0.885 − 1.94i)17-s + ⋯ |
| L(s) = 1 | + (1.53 − 0.988i)2-s + (−0.0821 + 0.571i)3-s + (0.972 − 2.13i)4-s + (0.429 − 0.125i)5-s + (0.438 + 0.960i)6-s + (−0.486 − 0.561i)7-s + (−0.349 − 2.42i)8-s + (−0.319 − 0.0939i)9-s + (0.535 − 0.617i)10-s + (1.04 + 0.668i)11-s + (1.13 + 0.730i)12-s + (−0.803 + 0.926i)13-s + (−1.30 − 0.382i)14-s + (0.0367 + 0.255i)15-s + (−1.40 − 1.61i)16-s + (−0.214 − 0.470i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37787 - 1.71630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37787 - 1.71630i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (1.69 + 4.48i)T \) |
| good | 2 | \( 1 + (-2.17 + 1.39i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (1.28 + 1.48i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.45 - 2.21i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.89 - 3.34i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.885 + 1.94i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (1.25 - 2.75i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.57 - 5.62i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.468 - 3.26i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.37 - 1.57i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.92 - 2.32i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.322 + 2.24i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + (-6.58 - 7.60i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.82 - 7.88i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.00 + 7.01i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (1.85 - 1.19i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (9.18 - 5.90i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.06 + 11.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.38 + 7.36i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (5.37 + 1.57i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.95 + 13.6i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.89 + 0.849i)T + (81.6 - 52.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.65798358714462690681771566683, −10.43208696731241958452903440435, −9.978761066247909688296230796334, −9.032865662228807596709333158537, −6.91159499746150496274583027477, −6.23273721007141130300434898049, −4.83094527864822771576554300258, −4.32475579039382231074421819866, −3.16923846598065930496448752297, −1.77464093880363117400541009207,
2.51635524980290571942235872961, 3.62300533468584094383644534238, 5.01997805895566297660030375809, 6.03943676796009048918489158877, 6.43321733042186896697170334473, 7.54270580473264659010880075685, 8.486007604481900905165088588852, 9.748299037864788795745901680014, 11.35729244660586412114389750724, 12.02111975660898299191004437363
